This application claims the priority of Japanese Patent Application No. 11-363241 filed on Dec. 21, 1999, which is incorporated herein by reference.
1. Field of the Invention
The present invention relates to a fringe analysis method using Fourier transform; and, in particular, to a fringe analysis method which can effectively use the Fourier transform method even when analyzing image data having a closed fringe pattern such as interference fringe pattern.
2. Description of the Prior Art
While the light wave interferometry has conventionally been known as an important means concerning precise measurement of object surfaces, there have been urgent demands for developing an interferometry technique (sub-fringe interferometry) which can read out information items smaller than a single stripe of interference fringe (one fringe) due to the necessity for measuring a surface or wavefront aberration at an accuracy of {fraction (1/10)} wavelength or higher.
For sub-fringe interferometry techniques, attention has been paid to those using the Fourier transform technique as described in xe2x80x9cBasics of sub-fringe interferometry,xe2x80x9d Kogaku, Vol. 13, No. 1, pp. 55-65, February, 1984.
However, though excellent in principle, the Fourier transform method leaves some problems unsolved, and have not always been effectively used in practice.
One of such problems lies in how to adapt the Fourier transform method to a closed fringe pattern.
Namely, while interference fringe pattern exhibits a pattern shaped like closed concentric circles when an object to be observed has a shape approximating a spherical or parabolic surface, it is difficult for a lower frequency signal component and a carrier frequency component (which is superimposed in the Fourier transform method since the surface to be observed and the reference surface are relatively tilted with each other) to be securely separated from each other in Fourier spectra plane when the Fourier transform method is used for such closed interference fringe pattern. Therefore, the original phase or wavefront can not be recovered (by the conventional Fourier transform method).
In view of the circumstances mentioned above, it is an object of the present invention to provide a fringe analysis method using Fourier transform which can securely separate the low frequency signal component and the carrier frequency component from each other in Fourier spectra plane when a fringe analysis is carried out for closed fringe image data by use of the Fourier transform method.
The present invention provides a fringe analysis method comprising the steps of converting fringe image data of a wavefront to be observed which are expressed by an orthogonal coordinate system into fringe image data represented by a different coordinate system; by using the conventional Fourier transform method, wavefront expressed in the converted coordinate system can be obtained; and then converting the obtained wavefront in the converted coordinate system back into a wavefront expressed in the original orthogonal coordinate system.
Preferably, the different coordinate system is a polar coordinate system.
Preferably, the fringe image data expressed by the orthogonal coordinate system comprise a closed fringe pattern, the origin of the polar coordinate system is set within the innermost of the closed fringe pattern.
Preferably, the origin is located near the center of the closed fringe pattern.
It will particularly be effective if the fringe analysis method using the Fourier transform in accordance with the present invention is applied to fringe image data carrying phase information and including carrier fringes superimposed thereon.
In particular, the present invention is effective in the case where the fringe image data are interference fringe image data, and carrier fringes caused by tilting the surface of the wavefront or the reference surface relatively with each other is superimposed on the interference fringe image data. Also, the present invention is considerably effective when the fringe image data are image data of moirxc3xa9 fringe pattern or other fringe pattern.
Preferably, ineffective data areas of the fringe image data represented by the converted coordinate system are filled by interpolation, and then the conventional Fourier transform method is applied to.
Preferably, the fringe image data represented by the converted coordinate system is multiplied by a predetermined window function.
Preferably, the fringe image data expressed by the orthogonal coordinate system is divided into a plurality of data areas, each of thus divided areas is converted into fringe image data represented by a converted coordinate system, conventional Fourier transform method is used so as to obtain wavefront corresponding to each area of the wavefront to be observed in the converted coordinate system, and then wavefronts of the plurality of areas represented by the converted coordinate system are combined together.
Preferably, the divided areas comprise a first angle area of 0xe2x89xa6xcex8 less than xcfx80 and a second angle area of xcfx80xe2x89xa6xcex82xcfx80.
It is also preferred that the divided areas comprise a first angle area of       0    ≤    θ     less than           π      2        ,
a second angle area of             π      2        ≤    θ     less than     π    ,
a third angle area of       π    ≤    θ     less than                   3        ⁢        π            2        ,
and a fourth angle area of             3      ⁢      π        2    ≤  θ   less than       2    ⁢          π      .      